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Coupling of Spin and Orbital Motion of Electrons in Carbon Nanotubes

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Coupling of Spin and Orbital Motion of Electrons in Carbon Nanotubes 1 Coupling of Spin and Orbital Motion of Electrons in Carbon Nanotubes F. Kuemmeth*, S. Ilani*, D. C. Ralph and P. L. McEuen Laboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca NY 14853 * These authors contributed equally to this work Electrons in atoms possess both spin and orbital degrees of freedom. In non-relativistic quantum mechanics, these are independent, resulting in large degeneracies in atomic spectra. However, relativistic effects couple the spin and orbital motion leading to the well- known fine structure in their spectra. The electronic states in defect-free carbon nanotubes (NTs) are widely believed to be four-fold degenerate1-10, due to independent spin and orbital symmetries, and to also possess electron-hole symmetry11. Here we report measurements demonstrating that in clean NTs the spin and orbital motion of electrons are coupled, thereby breaking all of these symmetries. This spin-orbit coupling is directly observed as a splitting of the four-fold degeneracy of a single electron in ultra-clean quantum dots. The coupling favours parallel alignment of the orbital and spin magnetic moments for electrons and anti-parallel alignment for holes. Our measurements are consistent with recent theories12,13 that predict the existence of spin-orbit coupling in curved graphene and describe it as a spin-dependent topological phase in NTs. Our findings have important implications for spin-based applications in carbon-based systems, entailing new design principles for the realization of qubits in NTs and providing a mechanism for all-electrical control of spins14 in NTs. Carbon-based systems are promising candidates for spin based applications such as spin- qubits14-19 and spintronics20-23 as they are believed to have exceptionally long spin coherence times due to weak spin-orbit interactions and the absence of nuclear spin in the 12C atom. Carbon NTs may play a particularly interesting role in this context because in addition to spin they offer a unique two-fold orbital degree of freedom that can also be used for quantum manipulation. The latter arises from the two equivalent dispersion cones (K and K’) in graphene, which lead to doubly-degenerate electronic orbits that encircle the nanotube circumference in a clockwise (CW) and counter-clockwise (CCW) fashion24 (Fig 1a). Together, the two-fold spin degeneracy and two-fold orbital degeneracy are generally assumed to yield a four-fold-degenerate electronic energy spectrum in clean NTs. Understanding the fundamental symmetries of this spectrum is at the heart of successful manipulation of these quantum degrees of freedom. A powerful way to probe the symmetries is by confining the carriers to a quantum dot 4,5,8,10,24,25 (QD) and applying a magnetic field parallel to the tube axis, B|| . The confinement creates bound states and the field interrogates their nature by coupling independently to their spin and orbital moments. In the absence of spin-orbit coupling, such a measurement should yield for a defect-free NT the energy spectrum shown in figure 1b. At B|| = 0 the NT spectrum should be four-fold degenerate. With increasing B|| the spectrum splits into pairs of CCW and CW states 2 (going down and up in energy respectively), each pair having a smaller internal spin splitting. Indications of approximate four-fold degeneracy have been observed in high-field measurements of electron addition spectra2-10 and inelastic cotunneling4,10 in nanotube QDs. However, in previous experiments disorder-induced splitting of the orbital degeneracy and electron-electron interactions in multi-electron QDs have masked the intrinsic symmetries at low energies. In this work we directly measure the intrinsic electronic spectrum by studying a single charge carrier, an electron or a hole, in an ultra-clean carbon nanotube QD. Remarkably, we find that the expected four-fold symmetry and electron-hole symmetry are broken by spin-orbit (SO) coupling, demonstrating that the spin and orbital motion in NTs are not independent degrees of freedom. The observed SO coupling further determines the filling order in the many-electron ground states, giving states quite different from models based purely on electron-electron interactions. The geometry of our devices is shown in Fig. 1c. A single small-bandgap NT is contacted by source and drain electrodes, and is gated from below by two gates (see methods). When biased, these gates shift the local Fermi energy in the NT thereby accumulating electrons or holes. In this work we use two independent gates to create a QD that is localized either above the left or above the right gate electrode. This is achieved by choosing appropriate combinations of gate voltages that pin the Fermi energy inside the gap on one side of the device while adding carriers to the other side (Fig 1c). Measurement of the linear conductance, G = dI / dVSD , through such a dot (Fig. 1e) shows Coulomb blockade peaks that correspond to the addition of individual carriers to the dot, and allows us to identify the first electron and first hole in the dot (see supplementary information for details). Having a single carrier in the dot enables us to study single-particle levels in the absence of electron-electron interactions, and thus to unambiguously identify the presence of spin-orbit coupling. The results reported here were observed in two independent devices and below we present data from one of them. We probe the quantum states of the NT using tunnelling spectroscopy. The differential conductance through the dot, G = dI / dVSD , is measured as a function of gate voltage, Vg , and source-drain bias, Vsd , as the first electron is added to the dot. Figure 2a shows a typical measurement taken at B|| = 300mT . The transition between the Coulomb blockade regions of zero and one electron features distinct resonances that correspond to the ground state (α) as well as the excited states (β, γ, δ) of the first electron. Their energies can be obtained from a line cut at constant Vsd (Fig 2b), by converting the gate voltages into energies (see methods). The magnetic field dependence of the one-electron states α,β, γ and δ is measured by taking Vg traces such as in Fig. 2b for different values of B|| . This is shown in Fig 2c, where we plot dI /dVSD as a function of Vg and B|| . The energies of the states α and β decrease with increasing B|| , hence we identify them as CCW orbital states. The states γ and δ increase in energy and are thus identified as CW orbital states. From the slopes of these resonances with respect to magnetic field we extract an orbital moment of µorb =1.55 meV/T and estimate the NT diameter to be d ≈ 5nm 24. 3 A striking difference is observed when we compare the measured excitation spectrum with the one predicted in Fig. 1b: At zero magnetic field the four states in our measurement are not degenerate but rather split into two pairs. To identify the nature of this splitting we note that with increasing magnetic field the energy difference between the states α and β increases while the difference between states γ and δ decreases, and both differences are consistent with a g-factor of an electron spin (Figure 2d). This observation allows us to identify unambiguously the spin and orbital composition of each energy level, as shown in the inset of Fig. 2c. At B|| = 0 the four-fold degeneracy is split into two Kramer doublets – the lower-energy doublet involves states with parallel alignment of orbital and spin magnetic moments, whereas the higher-energy doublet has states with anti-parallel alignment. The zero-field splitting is therefore identified as a spin-orbit splitting, with a value of ∆ SO = 0.37 ± 0.02 meV(extracted from Fig. 2d). At low fields (Figure 2e) the intersections of states with opposite spin directions (e.g. α and γ) show simple crossing, whereas states with parallel spin (e.g. β and γ) show avoided crossing, a signature of disorder-induced mixing between CCW and CW orbits ( ∆KK ' ). In previous experiments, the disorder-induced mixing was significantly larger, presumably obscuring the effects of SO coupling. In our measurements, the mixing is small, ∆ KK ' ≈ 65 µeV << ∆ SO , probably due to smooth electronic confinement, enabling the observation of SO effects. We further demonstrate the intrinsic nature of the effect by measuring identical excitation spectra for QDs formed at different locations along the same NT (Supp. Fig S1). Next, we show that SO coupling significantly affects the many-body ground states of multiple electrons in a QD. Figure 3a shows the magnetic field dependence of the addition energies for the N-electron ground states (N=-2 to +4), obtained by measuring the linear conductance as a function of Vg and B|| . Near zero magnetic field the sign of dVg / dB|| changes every time an electron is added (or removed), indicating that CCW and CW states are filled alternately. Similar addition sequences were explained in the past by repulsive electron-electron interactions driving electrons to occupy different orbits2-7,9,26 (Fig. 3b). However, in our nanotubes the underlying mechanism is entirely different. Comparing the one-electron excitation spectrum with the two electron-ground state (Fig. 3c), we see that the latter follows exactly the first excited state of the one-electron QD. Specifically, both start with a CW slope at low fields and flip to a CCW slope at the field associated with the SO splitting, B|| ≈ 125 mT . Thus the two-electron ground state is explained entirely by SO coupling (Fig. 3d). Note that below B|| ≈ 125 mT SO favours each of the two electrons to possess parallel orbital and spin moments, forcing them into two different orbital states.
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